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## G = S3×Dic3⋊C4order 288 = 25·32

### Direct product of S3 and Dic3⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — S3×Dic3⋊C4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×S3×Dic3 — S3×Dic3⋊C4
 Lower central C32 — C3×C6 — S3×Dic3⋊C4
 Upper central C1 — C22 — C2×C4

Generators and relations for S3×Dic3⋊C4
G = < a,b,c,d,e | a3=b2=c6=e4=1, d2=c3, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=c3d >

Subgroups: 618 in 197 conjugacy classes, 74 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×8], C22, C22 [×6], S3 [×4], C6 [×6], C6 [×7], C2×C4, C2×C4 [×13], C23, C32, Dic3 [×2], Dic3 [×11], C12 [×7], D6 [×6], C2×C6 [×2], C2×C6 [×7], C4⋊C4 [×4], C22×C4 [×3], C3×S3 [×4], C3×C6 [×3], C4×S3 [×8], C2×Dic3 [×3], C2×Dic3 [×12], C2×C12 [×2], C2×C12 [×6], C22×S3, C22×C6, C2×C4⋊C4, C3×Dic3 [×2], C3×Dic3 [×2], C3⋊Dic3 [×2], C3⋊Dic3, C3×C12, S3×C6 [×6], C62, Dic3⋊C4, Dic3⋊C4 [×6], C4⋊Dic3, C3×C4⋊C4, S3×C2×C4, S3×C2×C4 [×2], C22×Dic3 [×2], C22×C12, S3×Dic3 [×4], S3×Dic3 [×2], S3×C12 [×2], C6×Dic3 [×3], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6, S3×C4⋊C4, C2×Dic3⋊C4, Dic3⋊Dic3, C62.C22, C3×Dic3⋊C4, C6.Dic6, C2×S3×Dic3 [×2], S3×C2×C12, S3×Dic3⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×6], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, Dic6 [×2], C4×S3 [×4], C3⋊D4 [×2], C22×S3 [×2], C2×C4⋊C4, S32, Dic3⋊C4 [×4], C2×Dic6, S3×C2×C4 [×2], S3×D4, S3×Q8, C2×C3⋊D4, C2×S32, S3×C4⋊C4, C2×Dic3⋊C4, S3×Dic6, C4×S32, S3×C3⋊D4, S3×Dic3⋊C4

Smallest permutation representation of S3×Dic3⋊C4
On 96 points
Generators in S96
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 29 27)(26 30 28)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)(49 53 51)(50 54 52)(55 59 57)(56 60 58)(61 65 63)(62 66 64)(67 71 69)(68 72 70)(73 75 77)(74 76 78)(79 81 83)(80 82 84)(85 87 89)(86 88 90)(91 93 95)(92 94 96)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 49)(7 58)(8 59)(9 60)(10 55)(11 56)(12 57)(13 64)(14 65)(15 66)(16 61)(17 62)(18 63)(19 70)(20 71)(21 72)(22 67)(23 68)(24 69)(25 76)(26 77)(27 78)(28 73)(29 74)(30 75)(31 82)(32 83)(33 84)(34 79)(35 80)(36 81)(37 88)(38 89)(39 90)(40 85)(41 86)(42 87)(43 94)(44 95)(45 96)(46 91)(47 92)(48 93)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)(37 38 39 40 41 42)(43 44 45 46 47 48)(49 50 51 52 53 54)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)
(1 34 4 31)(2 33 5 36)(3 32 6 35)(7 26 10 29)(8 25 11 28)(9 30 12 27)(13 44 16 47)(14 43 17 46)(15 48 18 45)(19 38 22 41)(20 37 23 40)(21 42 24 39)(49 80 52 83)(50 79 53 82)(51 84 54 81)(55 74 58 77)(56 73 59 76)(57 78 60 75)(61 92 64 95)(62 91 65 94)(63 96 66 93)(67 86 70 89)(68 85 71 88)(69 90 72 87)
(1 23 11 17)(2 24 12 18)(3 19 7 13)(4 20 8 14)(5 21 9 15)(6 22 10 16)(25 46 31 40)(26 47 32 41)(27 48 33 42)(28 43 34 37)(29 44 35 38)(30 45 36 39)(49 67 55 61)(50 68 56 62)(51 69 57 63)(52 70 58 64)(53 71 59 65)(54 72 60 66)(73 94 79 88)(74 95 80 89)(75 96 81 90)(76 91 82 85)(77 92 83 86)(78 93 84 87)

G:=sub<Sym(96)| (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,21,23)(20,22,24)(25,29,27)(26,30,28)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,47,45)(44,48,46)(49,53,51)(50,54,52)(55,59,57)(56,60,58)(61,65,63)(62,66,64)(67,71,69)(68,72,70)(73,75,77)(74,76,78)(79,81,83)(80,82,84)(85,87,89)(86,88,90)(91,93,95)(92,94,96), (1,50)(2,51)(3,52)(4,53)(5,54)(6,49)(7,58)(8,59)(9,60)(10,55)(11,56)(12,57)(13,64)(14,65)(15,66)(16,61)(17,62)(18,63)(19,70)(20,71)(21,72)(22,67)(23,68)(24,69)(25,76)(26,77)(27,78)(28,73)(29,74)(30,75)(31,82)(32,83)(33,84)(34,79)(35,80)(36,81)(37,88)(38,89)(39,90)(40,85)(41,86)(42,87)(43,94)(44,95)(45,96)(46,91)(47,92)(48,93), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,34,4,31)(2,33,5,36)(3,32,6,35)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,44,16,47)(14,43,17,46)(15,48,18,45)(19,38,22,41)(20,37,23,40)(21,42,24,39)(49,80,52,83)(50,79,53,82)(51,84,54,81)(55,74,58,77)(56,73,59,76)(57,78,60,75)(61,92,64,95)(62,91,65,94)(63,96,66,93)(67,86,70,89)(68,85,71,88)(69,90,72,87), (1,23,11,17)(2,24,12,18)(3,19,7,13)(4,20,8,14)(5,21,9,15)(6,22,10,16)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,43,34,37)(29,44,35,38)(30,45,36,39)(49,67,55,61)(50,68,56,62)(51,69,57,63)(52,70,58,64)(53,71,59,65)(54,72,60,66)(73,94,79,88)(74,95,80,89)(75,96,81,90)(76,91,82,85)(77,92,83,86)(78,93,84,87)>;

G:=Group( (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,21,23)(20,22,24)(25,29,27)(26,30,28)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,47,45)(44,48,46)(49,53,51)(50,54,52)(55,59,57)(56,60,58)(61,65,63)(62,66,64)(67,71,69)(68,72,70)(73,75,77)(74,76,78)(79,81,83)(80,82,84)(85,87,89)(86,88,90)(91,93,95)(92,94,96), (1,50)(2,51)(3,52)(4,53)(5,54)(6,49)(7,58)(8,59)(9,60)(10,55)(11,56)(12,57)(13,64)(14,65)(15,66)(16,61)(17,62)(18,63)(19,70)(20,71)(21,72)(22,67)(23,68)(24,69)(25,76)(26,77)(27,78)(28,73)(29,74)(30,75)(31,82)(32,83)(33,84)(34,79)(35,80)(36,81)(37,88)(38,89)(39,90)(40,85)(41,86)(42,87)(43,94)(44,95)(45,96)(46,91)(47,92)(48,93), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,34,4,31)(2,33,5,36)(3,32,6,35)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,44,16,47)(14,43,17,46)(15,48,18,45)(19,38,22,41)(20,37,23,40)(21,42,24,39)(49,80,52,83)(50,79,53,82)(51,84,54,81)(55,74,58,77)(56,73,59,76)(57,78,60,75)(61,92,64,95)(62,91,65,94)(63,96,66,93)(67,86,70,89)(68,85,71,88)(69,90,72,87), (1,23,11,17)(2,24,12,18)(3,19,7,13)(4,20,8,14)(5,21,9,15)(6,22,10,16)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,43,34,37)(29,44,35,38)(30,45,36,39)(49,67,55,61)(50,68,56,62)(51,69,57,63)(52,70,58,64)(53,71,59,65)(54,72,60,66)(73,94,79,88)(74,95,80,89)(75,96,81,90)(76,91,82,85)(77,92,83,86)(78,93,84,87) );

G=PermutationGroup([(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,21,23),(20,22,24),(25,29,27),(26,30,28),(31,35,33),(32,36,34),(37,41,39),(38,42,40),(43,47,45),(44,48,46),(49,53,51),(50,54,52),(55,59,57),(56,60,58),(61,65,63),(62,66,64),(67,71,69),(68,72,70),(73,75,77),(74,76,78),(79,81,83),(80,82,84),(85,87,89),(86,88,90),(91,93,95),(92,94,96)], [(1,50),(2,51),(3,52),(4,53),(5,54),(6,49),(7,58),(8,59),(9,60),(10,55),(11,56),(12,57),(13,64),(14,65),(15,66),(16,61),(17,62),(18,63),(19,70),(20,71),(21,72),(22,67),(23,68),(24,69),(25,76),(26,77),(27,78),(28,73),(29,74),(30,75),(31,82),(32,83),(33,84),(34,79),(35,80),(36,81),(37,88),(38,89),(39,90),(40,85),(41,86),(42,87),(43,94),(44,95),(45,96),(46,91),(47,92),(48,93)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36),(37,38,39,40,41,42),(43,44,45,46,47,48),(49,50,51,52,53,54),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96)], [(1,34,4,31),(2,33,5,36),(3,32,6,35),(7,26,10,29),(8,25,11,28),(9,30,12,27),(13,44,16,47),(14,43,17,46),(15,48,18,45),(19,38,22,41),(20,37,23,40),(21,42,24,39),(49,80,52,83),(50,79,53,82),(51,84,54,81),(55,74,58,77),(56,73,59,76),(57,78,60,75),(61,92,64,95),(62,91,65,94),(63,96,66,93),(67,86,70,89),(68,85,71,88),(69,90,72,87)], [(1,23,11,17),(2,24,12,18),(3,19,7,13),(4,20,8,14),(5,21,9,15),(6,22,10,16),(25,46,31,40),(26,47,32,41),(27,48,33,42),(28,43,34,37),(29,44,35,38),(30,45,36,39),(49,67,55,61),(50,68,56,62),(51,69,57,63),(52,70,58,64),(53,71,59,65),(54,72,60,66),(73,94,79,88),(74,95,80,89),(75,96,81,90),(76,91,82,85),(77,92,83,86),(78,93,84,87)])

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C ··· 4H 4I 4J 4K 4L 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 12O 12P 12Q 12R order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 ··· 4 4 4 4 4 6 ··· 6 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 12 12 12 12 12 12 size 1 1 1 1 3 3 3 3 2 2 4 2 2 6 ··· 6 18 18 18 18 2 ··· 2 4 4 4 6 6 6 6 2 2 2 2 4 ··· 4 6 6 6 6 12 12 12 12

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + - + + + - + + - + - image C1 C2 C2 C2 C2 C2 C2 C4 S3 S3 D4 Q8 D6 D6 D6 C4×S3 Dic6 C4×S3 C3⋊D4 S32 S3×D4 S3×Q8 C2×S32 S3×Dic6 C4×S32 S3×C3⋊D4 kernel S3×Dic3⋊C4 Dic3⋊Dic3 C62.C22 C3×Dic3⋊C4 C6.Dic6 C2×S3×Dic3 S3×C2×C12 S3×Dic3 Dic3⋊C4 S3×C2×C4 S3×C6 S3×C6 C2×Dic3 C2×C12 C22×S3 Dic3 D6 D6 D6 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 2 1 8 1 1 2 2 3 2 1 4 4 4 4 1 1 1 1 2 2 2

Matrix representation of S3×Dic3⋊C4 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 5 11 0 0 0 0 0 8 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 12 3 0 0 0 0 8 1 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[5,0,0,0,0,0,11,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,8,0,0,0,0,3,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S3×Dic3⋊C4 in GAP, Magma, Sage, TeX

S_3\times {\rm Dic}_3\rtimes C_4
% in TeX

G:=Group("S3xDic3:C4");
// GroupNames label

G:=SmallGroup(288,524);
// by ID

G=gap.SmallGroup(288,524);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,58,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^6=e^4=1,d^2=c^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=c^3*d>;
// generators/relations

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